Chapter 4 Probabilities and Proportions Chapter 4 S

Chapter 4 Probabilities and Proportions Chapter 4, S 1

Chances of winning Lotto Chapter 4, S 2

Chances of winning Lotto Which one has the higher chance of winning? A. First B. Second C. Neither (same chance) Chapter 4, S 3

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Roulette In the casino I wait at the roulette wheel until I see a run of at least five reds in a row. I then bet heavily on a black. I am now more likely to win. Chapter 4, S 5

Roulette In the casino I wait at the roulette wheel until I see a run of at least five reds in a row. I then bet heavily on a black. I am now more likely to win. YES or NO? Chapter 4, S 6

Let’s Make a Deal Game Show Chapter 4, S 7

Let’s Make a Deal Game Show What is the better strategy? A. Switch B. Stay C. It makes no difference Chapter 4, S 8

Chapter 4 Probabilities and Proportions Chapter 4, S 9

What are probabilities? A probability is a number between 0 and 1 that quantifies uncertainty. 0 Impossible 1 Certain The probability that an event A occurs is written as pr(A ). Chapter 4, S 10

Examples: I toss a fair coin (where ‘fair’ means ‘equally likely outcomes’) § What are the possible outcomes? H & T § What is the probability it will turn up heads? 1/2 I choose a person at random and check which eye she/he winks with § What are the possible outcomes? L & R Chapter 4, S 11

Examples: What is the probability they I toss a fair coin (where ‘fair’ means ‘equally wink with their left eye? likely outcomes’) § § A. One-half What are the possible outcomes? H & T What is the probability it will turn up heads? 1/2 B. One-quarter C. Can’t tell I choose a person at random and check which eye she/he winks with § What are the possible outcomes? L & R Chapter 4, S 12

Examples: I toss a fair coin (where ‘fair’ means ‘equally likely outcomes’) § What are the possible outcomes? H & T § What is the probability it will turn up heads? 1/2 I choose a person at random and check which eye she/he winks with § What are the possible outcomes? L & R § What is the probability they wink with their left eye? ? Chapter 4, S 13

Equally likely outcomes For equally likely outcomes: pr(A ) = number of outcomes in A total number of outcomes The probability of getting a four when a fair dice is rolled is 1/6 Chapter 4, S 14

Probabilities and proportions are numerically equivalent. § The proportion of New Zealanders who are left handed is 0. 1. § A randomly selected New Zealander is left handed with a probability of 0. 1. Chapter 4, S 15

House Sales Weeks on the market Sale price Less than 3 3 – 6 More than 6 Total Under $300, 000 12 5 11 28 $300, 000 - $600, 000 70 69 47 186 Over $600, 000 52 47 30 129 Total 134 121 88 343 Let A be the event that a sale is made within 3 weeks B be the event that a sale is over $600, 000 Chapter 4, S 16

House Sales (a) Weeks on the market Sale price Less than 3 3 – 6 More than 6 Total Under $300, 000 12 5 11 28 $300, 000 - $600, 000 70 69 47 186 Over $600, 000 52 47 30 129 Total 134 121 88 343 What proportion of these sales were over $600, 000? pr(B ) = 129/343 = 0. 38 Chapter 4, S 17

House Sales (b) Weeks on the market Sale price Less than 3 3 – 6 More than 6 Total Under $300, 000 12 5 11 28 $300, 000 - $600, 000 70 69 47 186 Over $600, 000 52 47 30 129 Total 134 121 88 343 What proportion of these sales were not over $600, 000? pr(B ) = (28+186)/343 = 0. 62 Chapter 4, S 18

House Sales (c) Weeks on the market Sale price Less than 3 3 – 6 More than 6 Total Under $300, 000 12 5 11 28 $300, 000 - $600, 000 70 69 47 186 Over $600, 000 52 47 30 129 Total 134 121 88 343 What proportion of these sales were made in 3 or more weeks? pr(A ) = (121 + 88) / 343 = 0. 61 Chapter 4, S 19

House Sales (d) Weeks on the market Sale price Less than 3 3 – 6 More than 6 Total Under $300, 000 12 5 11 28 $300, 000 - $600, 000 70 69 47 186 Over $600, 000 52 47 30 129 Total 134 121 88 343 What proportion of these sales were made within 3 weeks and sold for over $600, 000? Chapter 4, S 20

House Sales (d) A. pr(A) B. pr(A and B) D. pr(A or B) C. pr(B) E. I don’t know Weeks on the market Sale price Less than 3 3 – 6 More than 6 Total Under $300, 000 12 5 11 28 $300, 000 - $600, 000 70 69 47 186 Over $600, 000 52 47 30 129 Total 134 121 88 343 What proportion of these sales were made within 3 weeks and sold for over $600, 000? Chapter 4, S 21

House Sales (d) A. 52/134 B. 52/343 D. 211/343 C. 52/129 E. I don’t know Weeks on the market Sale price Less than 3 3 – 6 More than 6 Total Under $300, 000 12 5 11 28 $300, 000 - $600, 000 70 69 47 186 Over $600, 000 52 47 30 129 Total 134 121 88 343 What proportion of these sales were made within 3 weeks and sold for over $600, 000? Chapter 4, S 22

House Sales (d) Weeks on the market Sale price Less than 3 3 – 6 More than 6 Total Under $300, 000 12 5 11 28 $300, 000 - $600, 000 70 69 47 186 Over $600, 000 52 47 30 129 Total 134 121 88 343 What proportion of these sales were made within 3 weeks and sold for over $600, 000? pr(A and B ) = 52/343 = 0. 15 Chapter 4, S 23

House Sales (e) Weeks on the market Sale price Less than 3 3 – 6 More than 6 Total Under $300, 000 12 5 11 28 $300, 000 - $600, 000 70 69 47 186 Over $600, 000 52 47 30 129 Total 134 121 88 343 What proportion of these sales were made within 3 weeks or sold for over $600, 000? pr(A or B ) = (134+129 -52)/343 = 0. 62 Chapter 4, S 24

House Sales (f) Weeks on the market Sale price Less than 3 3 – 6 More than 6 Total Under $300, 000 12 5 11 28 $300, 000 - $600, 000 70 69 47 186 Over $600, 000 52 47 30 129 Total 134 121 88 343 What proportion of these sales were on the market for less than 3 weeks given that they sold for over $600, 000? Chapter 4, S 25

House Sales (f) Weeks on the market Sale price Less than 3 3 – 6 More than 6 Total Under $300, 000 12 5 11 28 $300, 000 - $600, 000 70 69 47 186 Over $600, 000 52 47 30 129 Total 134 121 88 343 What proportion of these sales were on the market for less than 3 weeks given that they sold for over $600, 000? 52/129 = 0. 40 Chapter 4, S 26

Conditional Probabilities The sample space is reduced. Key words that indicate conditional probability are: given that, of those, if, assuming that “The probability of event A occurring given that event B has already occurred” is written in shorthand as: pr(A |B ) Chapter 4, S 27

House Sales (g) Weeks on the market Sale price Less than 3 3 – 6 More than 6 Total Under $300, 000 12 5 11 28 $300, 000 - $600, 000 70 69 47 186 Over $600, 000 52 47 30 129 Total 134 121 88 343 What proportion of the houses that sold in less than 3 weeks, sold for more than $600, 000? Chapter 4, S 28

The event in this question is? House Sales (g) A. Single know B. Joint C. Conditional Weeks on the market D. I don’t Sale price Less than 3 3 – 6 More than 6 Total Under $300, 000 12 5 11 28 $300, 000 - $600, 000 70 69 47 186 Over $600, 000 52 47 30 129 Total 134 121 88 343 What proportion of the houses that sold in less than 3 weeks, sold for more than $600, 000? Chapter 4, S 29

Conditional probability? House Sales (g) A. Yes B. No. C. I don’t know Weeks on the market Sale price Less than 3 3 – 6 More than 6 Total Under $300, 000 12 5 11 28 $300, 000 - $600, 000 70 69 47 186 Over $600, 000 52 47 30 129 Total 134 121 88 343 What proportion of the houses that sold in less than 3 weeks, sold for more than $600, 000? Chapter 4, S 30

House Sales (g) Weeks on the market Sale price Less than 3 3 – 6 More than 6 Total Under $300, 000 12 5 11 28 $300, 000 - $600, 000 70 69 47 186 Over $600, 000 52 47 30 129 Total 134 121 88 343 What proportion of the houses that sold in less than 3 weeks, sold for more than $600, 000? pr(B | A ) = 52/134 = 0. 39 Chapter 4, S 31

Filled jobs by industry and type (a) Type Industry Working owner Part time Full time Total Forestry & Mining 2 1 10 13 Electricity, Gas & Water 0 1 7 8 Total 132 458 1056 1646 What proportion of workers were part time employees? Chapter 4, S 32

Filled jobs by industry and type (a) Type Working owner Part time Full time Total Industry Forestry & Mining 2 1 10 13 Electricity, Gas & Water 0 1 7 8 Total 132 458 1056 1646 What proportion of workers were part time employees? The event in this question is? A. Single know B. Joint C. Conditional D. I don’t Chapter 4, S 33

Filled jobs by industry and type (a) Type Working owner Part time Full time Total Industry Forestry & Mining 2 1 10 13 Electricity, Gas & Water 0 1 7 8 Total 132 458 1056 1646 What proportion of workers were part time employees? Conditional probability? A. Yes B. No. C. I don’t know Chapter 4, S 34

Filled jobs by industry and type (a) Type Industry Working owner Part time Full time Total Forestry & Mining 2 1 10 13 Electricity, Gas & Water 0 1 7 8 Total 132 458 1056 1646 What proportion of workers were part time employees? pr(PT ) = 458/1646 = 0. 28 Chapter 4, S 35

Filled jobs by industry and type (b) Type Industry Accommodation, Cafes & Restaurants Working owner Part time Full time Total 10 57 33 99 The industry with the highest proportion of part time workers in March 2005 was accommodation, cafes & restaurants. What was this proportion? Chapter 4, S 36

Filled jobs by industry and type (b) The event in this question is? A. Single know Type C. Conditional B. Joint Industry Accommodation, Cafes & Restaurants D. I don’t Working owner Part time Full time Total 10 57 33 99 The industry with the highest proportion of part time workers in March 2005 was accommodation, cafes & restaurants. What was this proportion? Chapter 4, S 37

Filled jobs by industry and type (b) Conditional probability? A. Yes Type B. No. C. I don’t know Industry Accommodation, Cafes & Restaurants Working owner Part time Full time Total 10 57 33 99 The industry with the highest proportion of part time workers in March 2005 was accommodation, cafes & restaurants. What was this proportion? Chapter 4, S 38

Filled jobs by industry and type (b) Type Industry Accommodation, Cafes & Restaurants Working owner Part time Full time Total 10 57 33 99 The industry with the highest proportion of part time workers in March 2005 was accommodation, cafes & restaurants. What was this proportion? pr(PT |A ) = 57/99 = 0. 58 Chapter 4, S 39

Filled jobs by industry and type (c) Type Industry Working owner Part time Full time Total Retail Trade 30 90 112 232 Total 132 458 1056 1646 What proportion of workers were in the retail trade? Chapter 4, S 40

Filled jobs by industry and type (c) Type Industry Working owner Part time Full time Total Retail Trade 30 90 112 232 Total 132 458 1056 1646 What proportion of workers were in the retail trade? The event in this question is? A. Single know B. Joint C. Conditional D. I don’t Chapter 4, S 41

Filled jobs by industry and type (c) Type Industry Working owner Part time Full time Total Retail Trade 30 90 112 232 Total 132 458 1056 1646 What proportion of workers were in the retail trade? Conditional probability? A. Yes B. No. C. I don’t know Chapter 4, S 42

Filled jobs by industry and type (c) Type Industry Working owner Part time Full time Total Retail Trade 30 90 112 232 Total 132 458 1056 1646 What proportion of workers were in the retail trade? pr(R ) = 232/1646 = 0. 14 Chapter 4, S 43

Filled jobs by industry and type (d) Type Industry Working owner Part time Full time Total Education 2 37 87 125 Total 132 458 1056 1646 What proportion of workers were full time employees working in education? Chapter 4, S 44

Filled jobs by industry and type (d) Type Industry Working owner Part time Full time Total Education 2 37 87 125 Total 132 458 1056 1646 What proportion of workers were full time employees working in education? The event in this question is? A. Single B. Joint C. Conditional D. I don’t Chapter 4, S 45

Filled jobs by industry and type (d) Type Industry Working owner Part time Full time Total Education 2 37 87 125 Total 132 458 1056 1646 What proportion of workers were full time employees working in education? Conditional probability? A. Yes B. No. C. I don’t know Chapter 4, S 46

Filled jobs by industry and type (d) Type Industry Working owner Part time Full time Total Education 2 37 87 125 Total 132 458 1056 1646 What proportion of workers were full time employees working in education? pr(FT and E ) = 87/1646 = 0. 05 Chapter 4, S 47

Response Rates by Survey Format (a) Format Responses Nonresponses Total Paper only 325 1153 1478 Paper with web option 352 1116 1468 Web-only with response incentive 125 608 733 Web-only without response incentive 146 591 737 Total 948 3468 4416 What proportion of the students received an incentive and responded? 125/4416 = 0. 03 Chapter 4, S 48

Response Rates by Survey Format (b) Format Responses Nonresponses Total Paper only 325 1153 1478 Paper with web option 352 1116 1468 Web-only with response incentive 125 608 733 Web-only without response incentive 146 591 737 Total 948 3468 4416 What was the overall response rate to the survey? 948/4416 = 0. 21 Chapter 4, S 49

Response Rates by Survey Format (c) Format Responses Nonresponses Total Paper only 325 1153 1478 Paper with web option 352 1116 1468 Web-only with response incentive 125 608 733 Web-only without response incentive 146 591 737 Total 948 3468 4416 Which format had the highest response rate? Try it!!!! Chapter 4, S 50

Building a table from a story HIV Transmission A European study on the transmission of the HIV virus involved 305 heterosexual couples. Originally one of the partners in each couple was infected with the virus. There were 171 couples that always used condoms. From this group, 3 of the noninfected partners became infected with the What are virus. Of the 134 couples who did not the factors always use a condom, 16 of the nonof interest? infected partners became infected with the virus. Chapter 4, S 51

HIV Transmission Let C be the event that the couple always used condoms I be the event that the non-infected partner became infected Condom Usage Infection C C Total Status I I Total Chapter 4, S 52

HIV Transmission A European study on the transmission of the HIV virus involved 305 heterosexual couples. Condom Usage Infection C C Status Total I I Total 305 Chapter 4, S 53

HIV Transmission There were 171 couples that always used condoms. From this group, 3 of the non-infected partners became infected with the virus. Condom Usage Infection C C Status I Total 3 I Total 171 305 Chapter 4, S 54

HIV Transmission Of the 134 couples who did not always use a condom, 16 of the non-infected partners became infected with the virus. Condom Usage Infection C C Status I 3 16 171 134 Total I Total 305 Chapter 4, S 55

HIV Transmission Condom Usage Infection C C Status Total I 3 16 19 I 168 118 286 Total 171 134 305 Chapter 4, S 56

HIV Transmission (a) What proportion of the couples always used condoms? Condom Usage C Infection C Status Total I 3 16 19 I 168 118 286 Total 171 134 305 Chapter 4, S 57

HIV Transmission (a) What proportion of the couples always used condoms? The event in this question is? A. Single know B. Joint C. Conditional Condom Usage C Infection C Status D. I don’t Total I 3 16 19 I 168 118 286 Total 171 134 305 Chapter 4, S 58

HIV Transmission (a) What proportion of the couples always used condoms? A. 3/305 Condom Usage C Infection C Status B. 3/19 Total I 3 16 19 I 168 118 286 Total 171 134 305 C. 3/171 D. 171/305 E. Unsure Chapter 4, S 59

HIV Transmission (a) What proportion of the couples always used condoms? pr(C ) = 171/305 = 0. 56 Condom Usage C Infection C Status Total I 3 16 19 I 168 118 286 Total 171 134 305 Chapter 4, S 60

HIV Transmission (b) Of the couples who always used condoms, what proportion had a non-infected partner who became infected? Condom Usage C Infection C Status Total I 3 16 19 I 168 118 286 Total 171 134 305 Chapter 4, S 61

HIV Transmission (b) Of the couples who always used condoms, what proportion had a non-infected partner who became infected? Conditional probability? Condom Usage A. Yes B. No. C. I don’t know C Infection C Total Status I 3 16 19 I 168 118 286 Total 171 134 305 Chapter 4, S 62

HIV Transmission (b) Of the couples who always used condoms, what proportion had a non-infected partner who became infected? A. 3/305 Condom Usage C Infection C Status B. 3/19 Total I 3 16 19 I 168 118 286 Total 171 134 305 C. 3/171 D. 16/134 E. Unsure Chapter 4, S 63

HIV Transmission (b) Of the couples who always used condoms, what proportion had a non-infected partner who became infected? pr( I |C ) = 3/171 = 0. 02 Condom Usage C Infection C Status Total I 3 16 19 I 168 118 286 Total 171 134 305 Chapter 4, S 64

HIV Transmission (c) Of the couples who did not always use condoms, what proportion had a non-infected partner who became infected? Condom Usage C Infection C Status Total I 3 16 19 I 168 118 286 Total 171 134 305 Chapter 4, S 65

HIV Transmission (c) Of the couples who did not always use condoms, what proportion had a non-infected partner who became infected? Conditional probability? Condom Usage A. Yes B. No. C. I don’t know C Infection C Total Status I 3 16 19 I 168 118 286 Total 171 134 305 Chapter 4, S 66

HIV Transmission (c) Of the couples who did not always use condoms, what proportion had a non-infected partner who became infected? A. 16/19 Condom Usage C Infection C Status B. 16/305 Total I 3 16 19 I 168 118 286 Total 171 134 305 C. 118/286 D. 16/134 E. Unsure Chapter 4, S 67

HIV Transmission (c) Of the couples who did not always use condoms, what proportion had a non-infected partner who became infected? pr( I |C ) = 16/134 = 0. 12 Condom Usage C Infection C Status Total I 3 16 19 I 168 118 286 Total 171 134 305 Chapter 4, S 68

HIV Transmission (d) For a couple who did NOT always use a condom, how does the risk of the non-infected partner becoming infected compared to that for a couple who always used a condom? pr( I | C ) = 16/134 = 0. 12 pr( I | C ) = 3/171 = 0. 02 pr( I |C ) / pr( I |C ) Chapter 4, S 69

HIV Transmission (d) For a couple who did NOT always use a condom, how does the risk of the non-infected partner becoming infected compare to that for a couple who always used a condom? pr( I | C ) = 16/134 = 0. 12 pr( I | C ) = 3/171 = 0. 02 pr( I |C ) / pr( I |C ) = 0. 12/0. 02 = 6 times Chapter 4, S 70

Chances of Getting the Death Penalty In a study Radelet classified 326 murderers by race of the victim and type of sentence given to the murderer. 36 of the convicted murderers received the death sentence. Of this group, 30 had murdered a white person whereas 184 of the group that did not receive the death What are sentence had murdered a white person. the factors of interest? Chapter 4, S 71

Chances of Getting the Death Penalty Let W be the event that the victim is white D be the event that the sentence is death Sentence Victim’s Race W W Total D D Total Chapter 4, S 72

Chances of Getting the Death Penalty In a study Radelet classified 326 murderers by race of the victim and type of sentence given to the murderer. Sentence Victim’s Race W W Total D D Total 326 Chapter 4, S 73

Chances of Getting the Death Penalty 36 of the convicted murderers received the death sentence. Of this group, 30 had murdered a white person… Sentence D Victim’s Race W W 30 Total 36 D Total 326 Chapter 4, S 74

Chances of Getting the Death Penalty …whereas 184 of the group that did not receive the death sentence had murdered a white person. Sentence Victim’s Race W W Total D 30 6 36 D 184 106 290 Total 214 112 326 Chapter 4, S 75

Chances of Getting the Death Penalty (a) What is the probability of a murderer receiving the death sentence? Sentence Victim’s Race W W Total D 30 6 36 D 184 106 290 214 112 326 Total Chapter 4, S 76

Chances of Getting the Death Penalty (a) What is the probability of a murderer receiving the death sentence? The event in this question is? A. Single know B. Joint Sentence C. Conditional Victim’s Race W W Total D 30 6 36 D 184 106 290 214 112 326 Total D. I don’t Chapter 4, S 77

Chances of Getting the Death Penalty (a) What is the probability of a murderer receiving the death sentence? Conditional probability? A. Yes B. No. C. I don’t know Victim’s Race W Sentence W Total D 30 6 36 D 184 106 290 214 112 326 Total Chapter 4, S 78

Chances of Getting the Death Penalty (a) What is the probability of a murderer receiving the death sentence? Sentence Victim’s Race W W A. 214/326 Total B. 36/326 D 30 6 36 D 184 106 290 214 112 326 Total C. 30/36 D. 36/290 E. Unsure Chapter 4, S 79

Chances of Getting the Death Penalty (a) What is the probability of a murderer receiving the death sentence? pr(D ) = 36/326 = 0. 11 Sentence Victim’s Race W W Total D 30 6 36 D 184 106 290 214 112 326 Total Chapter 4, S 80

Chances of Getting the Death Penalty (b) What is the probability of a murderer receiving the death penalty given that the victim was white? Sentence Victim’s Race W W Total D 30 6 36 D 184 106 290 214 112 326 Total Chapter 4, S 81

Chances of Getting the Death Penalty (b) What is the probability of a murderer receiving the death penalty given that the victim was white? Conditional probability? A. Yes B. No. C. I don’t know Victim’s Race W Sentence W Total D 30 6 36 D 184 106 290 214 112 326 Total Chapter 4, S 82

Chances of Getting the Death Penalty (b) What is the probability of a murderer receiving the death penalty given that the victim was white? Sentence Victim’s Race W W A. 30/326 Total B. 30/36 D 30 6 36 D 184 106 290 214 112 326 Total C. 30/214 D. 184/214 E. Unsure Chapter 4, S 83

Chances of Getting the Death Penalty (b) What is the probability of a murderer receiving the death penalty given that the victim was white? pr(D |W ) = 30/214 = 0. 14 Sentence Victim’s Race W W Total D 30 6 36 D 184 106 290 214 112 326 Total Chapter 4, S 84

Chances of Getting the Death Penalty (c) What is the probability of a murderer receiving the death penalty given that the victim was black? Sentence Victim’s Race W W Total D 30 6 36 D 184 106 290 214 112 326 Total Chapter 4, S 85

Chances of Getting the Death Penalty (c) What is the probability of a murderer receiving the death penalty given that the victim was black? Conditional probability? A. Yes B. No. C. I don’t know Victim’s Race W Sentence W Total D 30 6 36 D 184 106 290 214 112 326 Total Chapter 4, S 86

Chances of Getting the Death Penalty (c) What is the probability of a murderer receiving the death penalty given that the victim was black? Sentence Victim’s Race W W A. 6/112 Total B. 6/36 C. 6/106 36 D. 6/326 D 30 6 D 184 106 290 214 112 326 Total E. Unsure Chapter 4, S 87

Chances of Getting the Death Penalty (c) What is the probability of a murderer receiving the death penalty given that the victim was black? pr(D |W ) = 6/112 = 0. 05 Sentence Victim’s Race W W Total D 30 6 36 D 184 106 290 214 112 326 Total Chapter 4, S 88

Chances of Getting the Death Penalty Michael Radelet, believed that, in Florida, the chance of getting the death penalty if you had killed a white person was three times the chance of getting the death penalty if you had killed a black person. pr(D |W ) = 0. 14 pr(D |W ) = 0. 05 Chapter 4, S 89

Raising EEO Issues 52. 5% of those surveyed were males. Of the males, 62% replied “Yes” and 13% replied “No”. Of the females, 55% replied “Yes” and 17% replied “No”. The remainder of both groups replied “Don’t know”. What are the factors of interest? Chapter 4, S 91

Raising EEO Issues 52. 5% of those surveyed were males. Response M Gender F Total Y N ? ? ? ? Total 52. 5/100 x 100000 47500 = 52500 100000 Chapter 4, S 92

Raising EEO Issues Of the males, 62% replied “Yes” and 13% replied “No”. The above percentages are Gender conditional statements: M F Response TRUE or FALSE? Total Y N ? Total 52500 47500 100000 Chapter 4, S 93

Raising EEO Issues Of the males, 62% replied “Yes” and 13% replied “No”. How many males replied “Yes”? Response M Y A. Gender 62, 000 F Total 47500 100000 B. 32, 550 C. 29, 450 D. Unsure N ? Total 52500 Chapter 4, S 94

Raising EEO Issues Of the males, 62% replied “Yes” and 13% replied “No”. Response Y N M Gender F Total 47500 100000 62% of 52500 = 32550 13% of 52500 = 6825 ? Total 52500 Chapter 4, S 95

Raising EEO Issues Of the females, 55% replied “Yes” and 17% replied “No”. Response Y N M Gender F Total 47500 100000 62% of 52500 = 32550 13% of 52500 = 6825 ? Total 52500 Chapter 4, S 96

Raising EEO Issues Of the females, 55% replied “Yes” and 17% replied “No”. The above percentages are Gender conditional statements: M F Response TRUE or FALSE? Y N Total 62% of 52500 = 32550 13% of 52500 = 6825 ? Total 52500 47500 100000 Chapter 4, S 97

Raising EEO Issues Of the females, 55% replied “Yes” and 17% replied “No”. How many females replied “Yes”? Gender A. 26, 125 M F Response B. 28, 875 62% of 52500 Y C. 47, 500 = 32550 13% of 52500 D. Unsure N = 6825 ? Total 52500 47500 Total 100000 Chapter 4, S 98

Raising EEO Issues Of the females, 55% replied “Yes” and 17% replied “No”. Response Y N M Gender 62% of 52500 = 32550 13% of 52500 = 6825 F Total 55% of 47500 = 26125 17% of 47500 = 8075 ? Total 52500 47500 100000 Chapter 4, S 99

Raising EEO Issues Of the females, 55% replied “Yes” and 17% replied “No”. Response Y N M Gender 62% of 52500 = 32550 13% of 52500 = 6825 F Total 55% of 47500 58675 = 26125 17% of 47500 14900 = 8075 ? Total 52500 47500 100000 Chapter 4, S 100

Raising EEO Issues The remainder of both groups replied “Don’t know”. Response M Gender ? 62% of 52500 = 32550 13% of 52500 = 6825 13125 Total 52500 Y N F Total 55% of 47500 58675 = 26125 17% of 47500 14900 = 8075 13300 26425 47500 100000 Chapter 4, S 101

Raising EEO Issues Of those who replied “No”, what proportion were female? Gender M F Total Y 32550 26125 58675 N 6825 8075 14900 ? 13125 13330 26425 Total 52500 47500 100000 Response Chapter 4, S 102

Raising EEO Issues Of those who replied “No”, what proportion were female? A. 0. 08 B. 0. 17 C. 0. 45 D. 0. 46 E. 0. 54 Gender M F Total Response Y 32550 26125 58675 N 6825 8075 14900 ? 13125 13330 26425 Total 52500 47500 100000 Chapter 4, S 103

Raising EEO Issues Of those who replied “No”, what proportion were female? Gender M F Total Y 32550 26125 58675 N 6825 8075 14900 ? 13125 13330 26425 Total 52500 47500 100000 Response Chapter 4, S 104

Raising EEO Issues Of those who replied “No”, what proportion were female? pr(F |N ) = 8075/14900 = 0. 54 Gender M F Total Y 32550 26125 58675 N 6825 8075 14900 ? 13125 13330 26425 Total 52500 47500 100000 Response Chapter 4, S 105

NZ Herald Tuesday 7 March 2006 “Last year, 183 people were diagnosed with HIV, . . . highest annual total since records began in 1985” “An estimated 1800 knowingly live with HIV (in NZ), but up to a third could have the virus and not know it. ” Chapter 4, S 106

NZ Herald Tuesday 20 th March 2009 “… 184 people were diagnosed with HIV, one more than the previous highest annual number of 183 in 2005. ” “… 152 people were infected through sexual contact, including 91 men through sex with other men, and 39 men and 22 women through heterosexual contact. ” Chapter 4, S 107

NZ Herald Monday 17 th March 2008 Incidence 2008: 184 2007: 156 2006: 177 2005: 183 Chapter 4, S 108

NZ Herald Monday 7 th March 2011 Record numbers of gay and bisexual men were diagnosed with HIV in New Zealand last year, new Incidence statistics show. 2010: 149 The AIDS Epidemiology Group at the University of Otago found 90 new infections of the virus which 2009: 151 can lead to AIDS among gay and bisexual men in 2008: 184 2010. This contrasted with the record low rates of new 2007: 156 infections among heterosexuals, who accounted 2006: 177 for just 35 of the 149 new HIV diagnoses made through antibody testing. 2005: 183 Chapter 4, S 109

AIDS NZ Newsletter March 2012 109 people were diagnosed with HIV through Incidence antibody testing in New Zealand in 2011: 109 59 were men infected through sex with other men, 28 (16 men and 12 women) through heterosexual 2010: 149 contact, one through injecting drug use, and one 2009: 151 child through mother-to-child transmission. For the remaining 20 people (15 men and 5 women) the 2008: 184 means of infection was unknown or information is still to be received. 2007: 156 2006: 177 2005: 183 Chapter 4, S 110

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Imperfect Testing: ELISA HIV Test For people who are: HIV positive: 99. 7% test positive pr(Test +ve| HIV+) = 0. 997 HIV negative: 0. 3% test positive (false positive) pr(Test +ve| HIV-) = 0. 003 Chapter 4, S 122

Imperfect Testing: ELISA HIV Test A person living in New Zealand, with a low HIV risk, has an ELISA HIV test. A positive test result occurs. What is the probability that the person has HIV? A. More than 80% B. Between 60% and 80% C. Between 40% and 60% D. Between 20% and 40% E. Less than 20% Chapter 4, S 123

Imperfect Testing: ELISA HIV Test For people who are: HIV positive: 99. 7% test positive pr(Test +ve| HIV+) = 0. 997 HIV negative: 0. 3% test positive (false positive) pr(Test +ve| HIV-) = 0. 003 It is estimated that 0. 1% of the New Zealand population are HIV positive. Chapter 4, S 124

ELISA HIV Test It is estimated that 0. 1% of the New Zealand population are HIV positive. Test Result HIV Status HIV+ HIV- Total Test +ve Test -ve Total 0. 1/100 x 1000000 =1000 999000 1000000 Chapter 4, S 125

ELISA HIV Test For people who are HIV positive: 99. 7% test positive Test Result HIV Status HIV+ HIV- Total Test +ve Test -ve Total 1000 999000 1000000 Chapter 4, S 126

ELISA HIV Test For people who are HIV positive: 99. 7% test positive Which one of the following statements HIV Status is true? This percentage given is: Test Result HIV+ HIVTotal A. Joint Test +ve B. Conditional on test result C. Conditional on HIV status Test -ve D. Unsure Total 1000 999000 1000000 Chapter 4, S 127

ELISA HIV Test For people who are HIV positive: 99. 7% test positive Test Result HIV Status HIV+ HIV- Test +ve 997 Test -ve 3 Total 1000 999000 Total 1000000 Chapter 4, S 128

ELISA HIV Test For people who are HIV negative: 0. 3% test positive (false positive) Test Result HIV Status HIV+ HIV- Test +ve 997 Test -ve 3 Total 1000 999000 Total 1000000 Chapter 4, S 129

ELISA HIV Test For people who are HIV negative: 0. 3% test positive (false positive) Test Result HIV Status HIV+ HIV- Test +ve 997 2997 Test -ve 3 996003 Total 1000 999000 Total 1000000 Chapter 4, S 130

ELISA HIV Test For people who are HIV negative: 0. 3% test positive (false positive) Test Result HIV Status HIV+ HIV- Total Test +ve 997 2997 3994 Test -ve 3 996006 Total 1000 999000 1000000 Chapter 4, S 131

ELISA HIV Test (a) Of those who test positive, what proportion are actually HIV+? HIV Status HIV+ HIV- Total Test +ve 997 2997 3994 Test -ve 3 996006 Total 1000 999000 1000000 Test Result Chapter 4, S 132

ELISA HIV Test (a) Of those who test positive, what proportion are actually HIV+? HIV Status Conditional probability? Test Result HIV+ HIVA. Yes B. No. C. I don’t know Total Test +ve 997 2997 3994 Test -ve 3 996006 Total 1000 999000 1000000 Chapter 4, S 133

ELISA HIV Test (a) Of those who test positive, what proportion are actually HIV+? Which one of the following statements HIV Status is true? This proportion described is: Test Result HIV+ Total HIVA. Joint Test +ve 997 2997 B. Conditional on test result 3994 C. Conditional on HIV status Test -ve 3 996006 D. Unsure Total 1000000 1000 999000 Chapter 4, S 134

ELISA HIV Test (a) Of those who test positive, what proportion are actually HIV+? pr(HIV+|Test +ve) = HIV Status Test Result HIV+ Total HIVTest +ve 997 2997 3994 Test -ve 3 996006 Total 1000 999000 1000000 Chapter 4, S 135

ELISA HIV Test (a) Of those who test positive, what proportion are actually HIV+? pr(HIV+|Test +ve) = 997/3994 = 0. 250 HIV Status Test Result HIV+ Total HIVTest +ve 997 2997 3994 Test -ve 3 996006 Total 1000 999000 1000000 Chapter 4, S 136

ELISA HIV Test (b) Of those who test positive, why are so few actually HIV+? HIV Status HIV+ HIV- Total Test +ve 997 2997 3994 Test -ve 3 996006 Total 1000 999000 1000000 Test Result Chapter 4, S 137

Imperfect Testing: ELISA HIV Test For people who are: HIV positive: 99. 7% test positive pr(Test +ve| HIV+) = 0. 997 HIV negative: 0. 3% test positive (false positive) pr(Test +ve| HIV-) = 0. 003 It is estimated that 0. 1% of the New Zealand population are HIV positive. Chapter 4, S 138

ELISA HIV Test (b) Of those who test positive, why are so few actually HIV+? HIV Status HIV+ HIV- Total Test +ve 997 2997 3994 Test -ve 3 996006 Total 1000 999000 1000000 Test Result Chapter 4, S 139

ELISA HIV Test (b) Of those who test positive, why are so few actually HIV+? HIV Status HIV+ HIV- Total Test +ve 997 2997 3994 Test -ve 3 996006 Total 1000 999000 1000000 Test Result Chapter 4, S 140

ELISA HIV Test (b) Of those who test positive, why are so few actually HIV+? Because of the very small number of HIV+ people overall. A very high percentage of a very small number (1000) gives a small number (997)! Chapter 4, S 141

ELISA HIV Test (c) In 1988 it was reported that an estimated 80% of drug addicts in New York City were HIV positive. HIV Status Test Result HIV+ HIVTotal Test +ve Test -ve Total Chapter 4, S 142

ELISA HIV Test (c) In 1988 it was reported that an estimated 80% of drug addicts in New York City were HIV positive. HIV Status Test Result HIV+ HIVTotal Test +ve Test -ve The event in this question is? Total A. Single B. Joint C. Conditional D. I don’t Chapter 4, S 143

ELISA HIV Test (c) In 1988 it was reported that an estimated 80% of drug addicts in New York City were HIV positive. HIV Status Test Result HIV+ HIVTotal Test +ve Test -ve Total 8000 2000 10000 Chapter 4, S 144

ELISA HIV Test (c) For people who are HIV positive: 99. 7% test positive Test Result HIV Status HIV+ HIV- Total Test +ve Test -ve Total 8000 2000 10000 Chapter 4, S 145

ELISA HIV Test (c) For people who are HIV positive: 99. 7% test positive Which one of the following statements HIV Status is true? This percentage given is: Test Result HIV+ HIVTotal A. Joint Test +ve B. Conditional on test result C. Conditional on HIV status Test -ve D. Unsure Total 8000 2000 10000 Chapter 4, S 146

ELISA HIV Test (c) For people who are HIV positive: 99. 7% test positive Test Result HIV Status HIV+ HIV- Test +ve 7976 Test -ve 24 Total 8000 2000 Total 10000 Chapter 4, S 147

ELISA HIV Test (c) For people who are HIV negative: 0. 3% test positive (false positive) HIV Status HIV+ HIV- Total Test +ve 7976 6 7982 Test -ve 24 1994 2018 Total 8000 2000 10000 Test Result Chapter 4, S 148

ELISA HIV Test (c) What is the probability that, in 1988, a randomly selected New York drug addict had HIV given that he/she tested positive? HIV Status HIV+ HIV- Total Test +ve 7976 6 7982 Test -ve 24 1994 2018 Total 8000 2000 10000 Test Result Chapter 4, S 149

ELISA HIV Test (c) What is the probability that, in 1988, a randomly selected New York drug addict had HIV given that he/she tested positive? Conditional probability? A. Yes Test Result B. No. C. I don’t know HIV Status HIV+ HIV- Total Test +ve 7976 6 7982 Test -ve 24 1994 2018 Total 8000 2000 10000 Chapter 4, S 150

ELISA HIV Test (c) What is the probability that, in 1988, a randomly selected New York drug addict had HIV given that he/she tested positive? Which one of the following statements is true? This proportion described is: HIV Status A. Joint Test Result HIV+ HIVTotal B. Conditional on test result Test +ve 7976 6 7982 C. Conditional on HIV status Test -ve 24 1994 2018 D. Unsure Total 8000 2000 10000 Chapter 4, S 151

ELISA HIV Test (c) What is the probability that, in 1988, a randomly selected New York drug addict had HIV given that he/she tested positive? pr(HIV+|Test +ve) = HIV Status HIV+ HIV- Total Test +ve 7976 6 7982 Test -ve 24 1994 2018 Total 8000 2000 10000 Test Result Chapter 4, S 152

ELISA HIV Test (c) What is the probability that, in 1988, a randomly selected New York drug addict had HIV given that he/she tested positive? pr(HIV+|Test +ve) = 7976/7982 = 0. 999 HIV Status HIV+ HIV- Total Test +ve 7976 6 7982 Test -ve 24 1994 2018 Total 8000 2000 10000 Test Result Chapter 4, S 153

Tax Audits Suppose that the incidence of tax evasion is 1 in 100 firms, that 90% of all cases of tax evasion are detected by an automated system and of those firms that are not evading tax, the system indicates that 5% are possibly evading tax. Chapter 4, S 154

Tax Audits Let T be the event the firm evades tax D be the event evasion is indicated by the system Test Result Tax Evasion Status T T Total D D Total 10000 Chapter 4, S 155

Tax Audits Suppose that the incidence of tax evasion is 1 in 100 firms… Test Result Tax Evasion Status T T Total D D Total 10000 Chapter 4, S 156

Tax Audits Suppose that the incidence of tax evasion is 1 in 100 firms… A. pr(T and D) = 1/100 B. pr(D|T) = 1/100 Tax Evasion Status C. pr(T) = 1/100 T T Test Result D. pr(D) = 1/100 D D Total E. pr(T|D) = 1/100 10000 Chapter 4, S 157

Tax Audits Suppose that the incidence of tax evasion is 1 in 100 firms… Test Result Tax Evasion Status T T Total D D Total 100 9900 10000 Chapter 4, S 158

Tax Audits …that 90% of all cases of tax evasion are detected by an automated system and of those firms that are not evading tax, the system indicates that 5% are possibly evading tax. Test Result Tax Evasion Status T T Total D D Total 100 9900 10000 Chapter 4, S 159

Tax Audits …that 90% of all cases of tax evasion are detected by an automated system and of those firms that are not evading tax, the system indicates that 5% are possibly evading tax. Which one of the following statements is true? The percentages stated are: Tax Evasion Status A. Joint Test Result T T Total D B. Conditional on tax evasion status D C. Conditional on test result Total. D. Unsure 100 9900 10000 Chapter 4, S 160

Tax Audits …that 90% of all cases of tax evasion are detected by an automated system and of those firms that are not evading tax, the system indicates that 5% are possibly evading tax. Test Result Tax Evasion Status T T Total D 90 495 585 D 10 9405 9415 Total 100 9900 10000 Chapter 4, S 161

Tax Audits Find the probability that a firm has actually evaded tax when the system indicates tax evasion. Test Result Tax Evasion Status T T Total D 90 495 585 D 10 9405 9415 Total 100 9900 10000 Chapter 4, S 162

Tax Audits Find the probability that a firm has actually evaded tax when the system indicates tax evasion. Conditional probability? A. Yes Test Result B. No. C. I don’t know Tax Evasion Status T T Total D 90 495 585 D 10 9405 9415 Total 100 9900 10000 Chapter 4, S 163

Tax Audits Find the probability that a firm has actually evaded tax when the system indicates tax evasion. A. 90/10000 B. 90/585 Test Result C. 90/100 Tax Evasion Status T T Total D 90 495 585 D 10 9405 9415 Total 100 9900 10000 Chapter 4, S 164

Tax Audits Find the probability that a firm has actually evaded tax when the system indicates tax evasion. pr(T|D) = Test Result Tax Evasion Status T T Total D 90 495 585 D 10 9405 9415 Total 100 9900 10000 Chapter 4, S 165

Tax Audits Find the probability that a firm has actually evaded tax when the system indicates tax evasion. pr(T|D) = 90/585 = 0. 15 Test Result Tax Evasion Status T T Total D 90 495 585 D 10 9405 9415 Total 100 9900 10000 Chapter 4, S 166

Statistical Independence Events A and B are statistically independent if pr(A |B ) = pr(A ) Chapter 4, S 167

Statistical Independence If A and B are statistically independent, then pr(A and B ) = pr(A ) × pr(B ) If the n events A 1, A 2, . . . An are mutually independent then pr(A 1 and A 2 and … and An ) = pr(A 1 ) × pr(A 2 ) × … × pr(An ) Chapter 4, S 168

People vs Collins Frequencies assumed by the Prosecution Yellow Car 1/10 Man with mustache 1/4 Girl with ponytail 1/10 Girl with blond hair Black man with beard Interracial couple in car 1/3 1/1000 pr(finding such a couple) = 1/10 1/4 1/10 1/3 1/1000 = 1/12, 000 !!!!! Chapter 4, S 169

The Sally Clark Story • Professional couple • Non-smoking environment pr(cot death) = 1/8, 500 pr(2 cot deaths) = 1/8, 500 x 1/8, 500 = 1/73, 000 Chapter 4, S 170

Probabilities, Meeting and Mating Chapter 4, S 171

Finding the Perfect Partner Essential Probability Between 25 and 45 1/2 Attractive, med height & weight 1/2 Very bright 1/25 Liberal 1/3 Relatively non-religious 1/3 Self-supporting 1/2 No kids 1/3 Chapter 4, S 172

Finding the Perfect Partner Essential Probability Funny, sense of humour Doesn’t drink or smoke Is not presently attached Cuddles, sexually assertive Likes me 1/3 1/4 1/2 1/10 Chapter 4, S 173

Finding the Perfect Partner Essential Probability Between 25 and 45 1/2 Attractive, medium height & weight 1/2 Very bright 1/25 Liberal 1/3 Relatively non-religious 1/3 Self-supporting 1/2 No kids 1/3 pr(finding perfect partner) = 1/25 … 1/3 1/10 = 1/2, 592, 000 !!!!! Chapter 4, S 174

Statistical Independence If A and B are statistically independent, then pr(A and B ) = pr(A ) × pr(B ) If the n events A 1, A 2, . . . An are mutually independent then pr(A 1 and A 2 and … and An ) = pr(A 1 ) × pr(A 2 ) × … × pr(An ) Chapter 4, S 175

Challenger Space Shuttle pr(one field joint ok) = 0. 977 pr(all 6 field joints ok) = pr(1 st ok) … pr(6 th ok) (by indep) = 0. 977 … 0. 977 = 0. 87 pr(system fails) = pr(at least 1 field joint fails) = 1 – pr(all 6 field joints ok) = 0. 13 Chapter 4, S 176

White Toyotas According to one New Zealand survey, 26% of cars are white and 27% of cars are made by Toyota. Now if these characteristics appear independently, and there is strong evidence that they do, the percentage of cars in New Zealand which are white Toyotas is: Chapter 4, S 177

White Toyotas According to one New Zealand survey, 26% of cars are white and 27% of cars are made by Toyota. Now if these characteristics appear independently, and there is strong evidence that they do, the percentage of cars in New Zealand which are white Toyotas is: A. pr(White and Toyota) B. pr(White | Toyota) C. pr(Toyota | White) D. Don’t know Chapter 4, S 178

White Toyotas According to one New Zealand survey, 26% of cars are white and 27% of cars are made by Toyota. Now if these characteristics appear independently, and there is strong evidence that they do, the percentage of cars in New Zealand which are white Toyotas is: pr(White and Toyota) = pr(White) × pr(Toyota) (by indep) = 0. 26 × 0. 27 = 7% Chapter 4, S 179